New limiting absorption and limit amplitude principles for periodic operators

Article

Abstract

We show a new limiting absorption and a new limit amplitude principle for periodic operators acting on functions defined on the whole \({\mathbb{R}^{d}}\) . For a differential operator \({\mathcal{L}}\) with periodic coefficients, we show the existence of the distributional limiting absorption solution of the Helmholtz equation (\({\mathcal{L}-\omega^{2}}\))v = g where \({\omega^{2}}\) is in the spectrum of \({\mathcal{L}}\) and regular in some sense. The limiting absorption solution is in some weighted L2-space with respect to the spatial variable. Moreover, we regard \({\omega}\) as an additional variable, and we take the limit also in a L2-norm with respect to \({\omega}\) . Based on this, we also derive a limit amplitude principle for the corresponding solution u of wave equation with time-harmonic right-hand side. In other words, we show in the same function spaces, the asymptotic behavior \({u(x) \sim e^{i\omega t}v^{-}(x)}\) as \({t \to +\infty}\) where v is the limiting absorption solution of the Helmholtz equation.

Mathematics Subject Classification (2010)

Primary 35B27 Secondary 78A40 35B40 35C99 

Keywords

Limiting absorption principle Limit amplitude principle Periodic operator Floquet–Bloch transform Helmholtz equation Wave equation Asymptotic behavior 

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© Springer Basel 2014

Authors and Affiliations

  1. 1.Institute for AnalysisKarlsruhe Institute of Technology (KIT)KarlsruheGermany

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